If y=sin−1x, show that (1−x2)dx2d2y−xdxdy=0.
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Step-by-Step Solution
Step 1: First Derivative
We are given the function y=sin−1x. To begin, we need to find the first derivative of y with respect to x. Recall that the derivative of sin−1x is 1−x21.
Step 2: Rearrange and Square
To simplify the process of finding the second derivative, we can rearrange the first derivative. Multiply both sides by 1−x2 and then square both sides to eliminate the square root. This gives us (1−x2)(dxdy)2=1.
Step 3: Second Derivative using Product Rule
Now, we differentiate the equation (1−x2)(dxdy)2=1 with respect to x. We use the product rule for the left side and the chain rule for (dxdy)2. The derivative of the right side (1) is 0.
Step 4: Simplify the Equation
We can factor out 2dxdy from both terms on the left side of the equation. This simplifies the expression and brings us closer to the desired form.
Step 5: Final Proof
Since dxdy=1−x21 is not equal to zero for x=±1, we can divide both sides of the equation by 2dxdy. This yields the required identity.